1. Topological insulators and superconductors
KITPC 2010 Shoucheng Zhang, Stanford University

2. Colloborators
Stanford group: Xiaoliang Qi, Andrei Bernevig, Congjun Wu, Chaoxing Liu, Taylor Hughes, Sri Raghu, Suk-bum Chung Stanford experimentalists: Yulin Chen, Ian Fisher, ZX Shen, Yi Cui, Aharon Kapitulnik, … Wuerzburg colleagues: Laurens Molenkamp, Hartmut Buhmann, Markus Koenig, Ewelina Hankiewicz, Bjoern Trauzettle IOP colleagues: Zhong Fang, Xi Dai, Haijun Zhang, … Tsinghua colleagues: Qikun Xue, Jinfeng Jia, Xi Chen,…

3. Outline
Models and materials of topological insulators General theory of topological insulators, exotic particles Topological superconductors

4. The search for new states of matter
The search for new elements led to a golden age of chemistry.
The search for new particles led to the golden age of particle physics.
In condensed matter physics, we ask what are the fundamental states of matter?
In the classical world we have solid, liquid and gas. The same H2O molecules can condense into ice, water or vapor.
In the quantum world we have metals, insulators, superconductors, magnets etc.
Most of these states are differentiated by the broken symmetry.
Magnet: Broken rotational symmetry
Superconductor: Broken gauge symmetry
Crystal: Broken translational symmetry

5. The quantum Hall state, a topologically non-trivial state of matter
TKNN integer=the first Chern number.
Topological states of matter are defined and described by topological field theory:
von Klitzing, 1980
Physically measurable topological properties are all contained in the topological field theory, e.g. QHE, fractional charge, fractional statistics etc…

6. Discovery of the 2D and 3D topological insulator
HgTe Theory: Bernevig, Hughes and Zhang, Science 314, 1757 (2006) Experiment: Koenig et al, Science 318, 766 (2007) BiSb Theory: Fu and Kane, PRB 76, (2007) Experiment: Hsieh et al, Nature 452, 907 (2008) Bi2Te3, Sb2Te3, Bi2Se3 Theory: Zhang et al, Nature Physics 5, 438 (2009) Experiment Bi2Se3: Xia et al, Nature Physics 5, 398 (2009), Experiment BieTe3: Chen et al Science 325, 178 (2009) On average 2-3 paper per day on the subject!

7. Topological Insulator is a New State of Quantum Matter

8. From traffic jam to info-superhighway on chip
Traffic jam inside chips today
Info highways for the chips in the future

9. Quantum Hall effect and quantum spin Hall effect

10. Topological protection (Qi and Zhang, Phys Today, Jan, 2010)
Spin=1/2
y=>-y

11. Chiral (QHE) and helical (QSHE) liquids in D=1k kF
-kF k kF
-kF
The QHE state spatially separates the two
chiral states of a spinless 1D liquid
The QSHE state spatially separates the
four chiral states of a spinful 1D liquid x
2=1+1
4=2+2 x
Benervig and Zhang, Kane and Mele

12. Time reversal symmetry in quantum mechanics
Wave function of a particle with integer spin changes by 1 under 2p rotation.
Spin=1
Wave function of a half-integer spin changes by -1 under 2p rotation.
Kramers theorem, in a time reversal invariant system with half-integer spins, T2=-1, all states for degenerate doublets.
y=> y
Application in condensed matter physics: Anderson’s theorem. BCS pair=(k,up)+(-k,down). General pairing between Kramers doublets.
Spin=1/2
y=>-y

13. Quantum protection by T2=-1 (Qi&Zhang, Phys Today, Jan, 2010)
Transport experiments:
Molenkamp group
STM experiments:
Yazdani group, Kapitulnik group, Xue group

14. Kane and Mele, Wu, Bernevig and Zhang; Xu and Moore
The topological distinction between a conventional insulator and a QSH insulator
Kane and Mele, Wu, Bernevig and Zhang; Xu and Moore
Band diagram of a conventional insulator, a conventional insulator with accidental surface states (with animation), a QSH insulator (with animation). Blue and red color code for up and down spins. e k
k=0 or p
Trivial
Trivial
Non-trivial

15. Quantum mechanics and special relativity:
Spin=1/2
Quantum mechanics predicts spin ½ particles
Relativity predicts spin-orbit coupling

17. Band Structure of HgTe S
P1/2
P3/2 S P S
P3/2
P1/2 S P

18. Band inversion in HgTe leads to a topological quantum phase transition
Let us focus on E1, H1
bands close to
crossing point
HgTe
HgTe E1 H1
CdTe
CdTe
CdTe
CdTe H1 E1
normal
inverted

19. The model of the 2D topological insulator (BHZ, Science 2006)
Square lattice with 4-orbitals per site:
Nearest neighbor hopping integrals. Mixing matrix elements between the s and the p states must be odd in k.
Similar to relativistic Dirac equation in 2+1 dimensions, with a mass term tunable by the sample thickness d! m/B<0 for d>dc.

20. Mass domain wall
Cutting the Hall bar along the y-direction we see a domain-wall structure in the
band structure mass term. This leads to states localized on the domain wall which still disperse along the x-direction, similar to Jackiw-Rebbi soliton. y y x
m/B<0 m x kx E
Bulk E

21. Experimental setup
High mobility samples of HgTe/CdTe quantum wells have been fabricated.
Because of the small band gap, about several meV, one can gate dope this system from n to p doped regimes.
Two tuning parameters, the thickness d of the quantum well, and the gate voltage.
(Koenig et al, Science 2007)

22. Experimental observation of the QSH edge state (Konig et al, Science 2007)

23. Nonlocal transport in the QSH regime, (Roth et al Science 2009)
R14,14=3/4 h/e2 1 3 2 4
I: 1-4
V: 2-3
R14,23=1/4 h/e2 23

24. No QSH in graphene
Bond current model on honeycomb lattice (Haldane, PRL 1988).
Spin Hall insulator (Murakami, Nagaosa and Zhang 2004)
Spin-orbit coupling in graphene (Kane and Mele 2005). They took atomic spin-orbit coupling of 5meV to estimate the size of the gap.
Yao et al, Min et al 2006 showed that the actual spin-orbit gap is given by D2so/Dps =10-3 meV.
Similar to the seasaw mechanism of the neutrino mass in the Standard Model!

25. Single valley 2D Dirac cone
At the critical thickness, the BHZ model reduces to the massless Dirac model in 2+1d, with a single valley.
Recent experiments on HgTe has reached this critical point!
HgTe provides an ideal platform to study 2D massless Dirac fermions!

26. 3D insulators with a single Dirac cone on the surface
(b) z
(a) y x y x
Quintuple
layer
(c) C A t2 B t3 t1 C
Se2 Bi
Se1 A B C

27. Relevant orbitals of Bi2Se3 and the band inversion
0.6 Bi
E (eV)
0.2 Se c
-0.2
0.2
0.4 
(eV)
(I)
(II)
(III)

28. (a) Sb2Se3
(b) Sb2Te3
(c) Bi2Se3
(d) Bi2Te3

29. Pz+, up, Pz-, up, Pz+, down, Pz-, down
Model for topological insulator Bi2Te3, (Zhang et al, 2009)
Pz+, up, Pz-, up, Pz+, down, Pz-, down
Single Dirac cone on the surface of Bi2Te3
Surface of Bi2Te3 = ¼ Graphene !

30. Oscillatory crossover from 3D to 2D (Liu et al, 0908.3654)

31. Doping evolution of the FS and band structure
Arpes experiment on Bi2Te3 surface states, Shen group
Doping evolution of the FS and band structure
EF(undoped)
BCB bottom
Dirac point position
Undoped
Under-doped
Optimally-
doped
Over-doped
BVB bottom

32. Arpes experiment on Bi2Se3 surface states, Hasan group

33. Surface states of TIs vs Rashba SOC
3D TI
Surface states of TIs vs Rashba SOC
Unlike graphene, here two components are related by time reversal, and Pauli matrix  is the real spin.
Surface Rashba term: Breaking of inversion symmetry at the surface.
Conduction band
Conduction band E E
Strong SOC
P: strong SOC drive the system into non-trivial state.
Valence band
Valence band

34. Spin-plasmon collective mode (Raghu, Chung, Qi+SCZ, PRL2009)
General operator identity:
Density-spin coupling:

35. AB oscillations in nano-ribbon of Bi2Se3, Cui group

36. STM experiments on quasi-particle interference, Yazdani, Kapitulnik, Xue groups, theory Lee et al

37. Surface Landau levels, Xue group

38. Completing the table of Hall effects
1879
Anomalous Hall
1889
Spin Hall
2004
QHE
1980
QAHE
2010?
QSHE
2006/2007

39. New mechanism for ferromagnetic order in insulating Bi2-xTMxSe3, TM=Fe, Cr: cond-mat/1002.0946

40. Theoretical prediction of the quantized AHE state

41. Theoretical prediction of the quantized AHE state
Qi, Wu & Zhang, PRB74, (2006)
Liu et al, PRL, 101, (2008)
Yu et al, cond-mat/

42. Theoretical prediction of the quantized AHE state

43. General theory of topological insulators
Topological field theory of topological insulators. Generally valid for interacting and disordered systems. Directly measurable physically. Relates to axion physics! (Qi, Hughes and Zhang)
For a periodic system, the system is time reversal symmetric only when
q=0 => trivial insulator
q=p => non-trivial insulator
Topological band theory based on Z2 topological band invariant of single particle states.
(Fu, Kane and Mele, Moore and Balents, Roy)

44. E M j// q term with open boundaries
q=p implies QHE on the boundary with
For a sample with boundary, it is only insulating when a small T-breaking field is applied to the boundary. The surface theory is a CS term, describing the half QH.
Each Dirac cone contributes sxy=1/2e2/h to the QH. Therefore, q=p implies an odd number of Dirac cones on the surface!
T breaking E M
j//
Surface of a TI = ¼ graphene

45. Generalization of the QH topology state in d=2 to time reversal invariant topological state in d>2, in Science 2001

46. The periodic table of topological states:
(Qi, Hughes and Zhang, Ludwig et al, Kitaev)
The TRI topological insulators form a dimensional chain: 4D=>3D=>2D

47. The 4D TRI topological insulator directly inspired the discovery of the intrinsic spin Hall effect in d=3, Science 2003
“Recently, the QHE has been generalized to four spatial dimensions (4). In that case, an electric field induces an SU(2) spin current through the nondissipative transport equation…The quantum Hall response in that system is physically realized though the spin-orbit coupling in a time reversal symmetric system.”

48. TRB topological insulators in d=2
Chern-Simons topological field theory, odd under TR:
1st Chern number

49. TRI topological insulators in d=4
Chern-Simons topological field theory, even under TR:
If we perform Kaluza-Klein compactification, we obtain the effective field theory of the d=3 topological insulator!
2st Chern number
If we replace k4 by an adiabatic parameter, we obtain the quantized cyclic change of the magneto-electric polarization in d=3!
The best way to understand TRI TI is D=4 => D=3 => D=2
The best way to understand critical phenomenon is D=4 => D=4-e

50. q Dimensional reduction A5 x5 (x, y, z)
From 4D QHE to the 3D topological insulator
Zhang & Hu, Qi, Hughes & Zhang A5 x5 q
(x, y, z)
From 3D axion action to the 2D QSH
Goldstone & Wilzcek

51. TRI topological insulators in d=3
Axion field theory, even under TR only when q=0, p:
Magneto-electric polarization (QHZ 2008):

52. Generalization to general interacting TI (Wang, Qi, SCZ)
Topological order parameter for generally interacting TI
Experimentally measurable through the topological magneto-electric effect
WZW extension u introduces integer ambiguity of P3
P3 is topologically quantized to be integer or half-integer
Also applies to disordered systems, see Li et al, Groth et al.

53. Topological field theory and the family tree
Topological field theory of the QHE: (Thouless et al, Zhang, Hansson and Kivelson)
Topological field theory of the TI: (Qi, Hughes and Zhang, 2008)
More extensive and general classification soon followed (Kitaev, Ludwig et al)

54. TRI topological insulator d=2, characterized by the discrete Z2 topological number in 2005
Topological band theory

55. The continuum TI formula and the discrete TI formula are pre-destined to converge! (Wang, Qi and Zhang, )

56. Equivalence between the integral and the discrete topological invariants (Wang, Qi and Zhang, )
LHS=QHZ definition of TI, RHS=FKM definition of TI

57. Time Reversal Invariant Topological Insulators
Zhang & Hu 2001: TRI topological insulator in D=4. => Root state of all TRI topological insulators.
Murakami, Nagaosa & Zhang 2004: Spin Hall insulator with spin-orbit coupled band structure.
Kane and Mele, Bernevig and Zhang 2005: Quantum spin Hall insulator with and without Landau levels.
Fu, Kane & Mele, Moore and Balents, Roy 2007: Topological band theory based on Z2
Qi, Hughes and Zhang 2008: Topological field theory based on F F dual.
TRB Chern-Simons term in D=2: A0=even, Ai=odd
TRI Chern-Simons term in D=4: A0=even, Ai=odd

58. Equivalence between the integral and the discrete topological invariants (Wang, Qi and Zhang, )
LHS=QHZ definition of TI, RHS=FKM definition of TI

59. Topological field theory of QHE and TI
Topological field theory of the QHE: (Thouless et al, Zhang, Hansson and Kivelson)
Topological field theory of the TI: (Qi, Hughes and Zhang, 2008)
Family tree of TI:
Root state of TI=4DQHE (Zhang+Hu, Science, 2001)

60. RKKY coupling of the surface states (Liu et al, PRL 2009)

61. Low frequency Faraday/Kerr rotation
(Qi, Hughes and Zhang, PRB78, , 2008, Zhang group 2010, MacDonald group 2010)
Adiabatic
Requirement:
(surface gap)
Universal quantization in units of the fine structure constant!

62. Seeing the magnetic monopole thru the mirror of a TME insulator, (Qi et al, Science 323, 1184, 2009)
higher order feed back
(for =’, =’)
similar to Witten’s dyon effect
Magnitude of B:

63. Dynamic axions in topological magnetic insulators
(Li et al, Nature Physics 2010)
Hubbard interactions leads to anti-ferromagnetic order
Effective action for dynamical axion

64. Axions and dark matter on your desktop
Axions and dark matter on your desktop? (Zhang group, Nature Physics 2010)
Now Shou-Cheng Zhang and his colleagues inform us that, all along, axions have been lurking unrecognized on surfaces
of topological insulators.
Frank Wilzcek, NATURE 458, 129 (2009)

65. An electron-monopole dyon becomes an anyon!

66. Topological Mott insulators
Dynamic generation of spin-orbit coupling can give rise to TMI (Raghu et al, PRL 2008).
Interplay between spin-orbit coupling and Mott physics in 5d transition metal Ir oxides, Nagaosa, SCZ et al PRL 2009, Balents et al, Franz et al.
Topological Kondo insulators (Coleman et al)

67. Qi, Hughes, Raghu and Zhang, PRL, 2009
Topological insulators and superconductors
Full pairing gap in the bulk, gapless Majorana edge and surface states
Chiral Majorana fermions
Chiral fermions
massless Majorana fermions
massless Dirac fermions
Qi, Hughes, Raghu and Zhang, PRL, 2009

68. Topological superconductors and superfluids
The BCS-BdG model for 2D equal spin pairing  model of 2D TI by BHZ
where p+=px+ipy. The edge Hamiltonian is given by:
forming a pair of Majorana fermions. Mass term breaks T symmetry=> topological protection!
Qi, Hughes, Raghu and Zhang, PRL, 2009
Schnyder et al, PRB, 2008
Kitaev
Roy
Tanaka, Nagaosa et al, PRB, 2009
Sato, PRB, 2009

69. Probing He3B as a topological superfluid (Chung and Zhang, 2009)
The BCS-BdG model for He3B  Model of the 3D TI by Zhang et al
Surface Majorana state:
Qi, Hughes, Raghu and Zhang, PRL, 2009
Schnyder et al, PRB, 2008
Kitaev
Roy

70. Chiral superconductor obtained from QAH+SC (Qi, Hughes and SCZ, 2010)

71. From geometrical to topological laws of physics
Albert Einstein: fundamental laws of physics are laws of geometry.
Indeed, the fundamental field equations of the Standard Model, Einstein, Maxwell, Yang-Mills are all geometrical field equations.
What about topological field equations?
The only topological term within the Standard Model:
This term defines and described the TI
Frank Wilzcek: Topological insulator is a window into the universe! (Nature 458, 129, 2009)

72. General reading
For a video introduction of topological insulators and superconductors, see

73. Summary: discovery of new states of matter
s-wave superconductor
Crystal
Magnet
Topological insulators
Quantum Spin Hall